Bott vanishing for Fano threefolds

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that \(H^j(X,\Omega ^i_X\otimes L)=0\) $H^j(X,{\mathrm{\Omega }}_X^i\otimes L)=0$ for \(j>0\) $j>0$ , \(i\ge 0\) $i\ge 0$ , and L ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected.